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Hausdorff measure of critical set for Luzin $N$ condition

Anna Doležalová, Marika Hrubešová, Tomáš Roskovec

TL;DR

This work investigates how large the critical set where the Luzin $N$ condition fails can be, within generalized gauge-function Hausdorff measures. It leverages a Ponomarev-type Cantor construction to build Sobolev or grand-Sobolev homeomorphisms $f$ that map a Lebesgue-null Cantor-type set $C_A$ to a positive-measure set $C_B$, while precisely controlling the size of $C_A$ in terms of a gauge function $h$. The authors prove two key theorems showing how $\,\mathcal{H}^h(C_A)$ can be arranged to be strictly between zero and infinity or to vanish, all while maintaining $f\in W^{1,n)}$ and $J_f>0$ almost everywhere, thereby illustrating sharp gauge-scale behavior of the Luzin $N$ condition. These results extend the classical Luzin theory by highlighting how the exceptional set size can be tuned across a wide range of Hausdorff scales, with implications for area, coarea, and distortion theories in low-regularity mappings.

Abstract

It is well-known that there is a Sobolev homeomorphism $f\in W^{1,p}([-1,1]^n,[-1,1]^n)$ for any $p<n$ which maps a set $C$ of zero Lebesgue $n$-dimensional measure onto the set of positive measure. We study the size of this critical set $C$ and characterize its lower and upper bounds from the perspective of Hausdorff measures defined by a general gauge function.

Hausdorff measure of critical set for Luzin $N$ condition

TL;DR

This work investigates how large the critical set where the Luzin condition fails can be, within generalized gauge-function Hausdorff measures. It leverages a Ponomarev-type Cantor construction to build Sobolev or grand-Sobolev homeomorphisms that map a Lebesgue-null Cantor-type set to a positive-measure set , while precisely controlling the size of in terms of a gauge function . The authors prove two key theorems showing how can be arranged to be strictly between zero and infinity or to vanish, all while maintaining and almost everywhere, thereby illustrating sharp gauge-scale behavior of the Luzin condition. These results extend the classical Luzin theory by highlighting how the exceptional set size can be tuned across a wide range of Hausdorff scales, with implications for area, coarea, and distortion theories in low-regularity mappings.

Abstract

It is well-known that there is a Sobolev homeomorphism for any which maps a set of zero Lebesgue -dimensional measure onto the set of positive measure. We study the size of this critical set and characterize its lower and upper bounds from the perspective of Hausdorff measures defined by a general gauge function.

Paper Structure

This paper contains 7 sections, 2 theorems, 29 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Motivation and history
  3. Ponomarev construction and its refinements
  4. Further applications of the Luzin N condition and related questions
  5. Preliminaries
  6. Ponomarev construction
  7. Proof of Theorem \ref{['th:velkyponomarev']} and Theorem \ref{['th:malyponomarev']}

Key Result

Theorem 1.1

Let $Q_0=[-1,1]^n$, $\tau:(0,\infty)\to [1,\infty)$ be a monotone, continuous function such that $\lim_{t\to 0+}\tau(t)=\infty$ and for all $p\in (0,1]$ there exists $x_p \in(0,1)$ such that for all $t\in (0,x_p)$ we have Let $h(t):[0,\infty)\to[0,\infty)$ be a gauge function, i.e., continuous non-decreasing function such that $h(0)=0$, and satisfying $h(t)=t^n\tau(t)$ on $(0,\infty)$. Then there

Figures (2)

  • Figure 1: First two steps in Ponomarev construction of $f$. Above: $f_1$ maps $\bigcup_{{\mathbf{v}}\in{\mathbb{V}}} Q_{\mathbf{v}}$ onto $\bigcup_{{\mathbf{v}}\in{\mathbb{V}}} \tilde{Q}_{\mathbf{v}}$. Below: $f_2$ maps $\bigcup_{{\mathbf{v}}\in{\mathbb{V}^2}} Q_{\mathbf{v}}$ onto $\bigcup_{{\mathbf{v}}\in{\mathbb{V}^2}} \tilde{Q}_{\mathbf{v}}$.
  • Figure 2: Mapping $f_k$ transforms $Q_{\mathbf{v}}$ onto $\tilde{Q}_{\mathbf{v}}$ (the gray area) and $Q'_{\mathbf{v}}\setminus Q_{\mathbf{v}}$ onto $\tilde{Q}'_{\mathbf{v}}\setminus \tilde{Q}_{\mathbf{v}}$ (the white area), ${\mathbf{v}}\in{\mathbb{V}}^k$.

Theorems & Definitions (5)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 3.1
  • proof : Proof of Theorem \ref{['th:malyponomarev']}
  • proof : Proof of Theorem \ref{['th:velkyponomarev']}